flat area to sphere or turn to half of sphere
To sphere is actually the correct way if you do not want to use proportional editing, but you have to make sure that yout pivot center is set to where you want the center of the sphere to be.
The easiest way is probably to set the pivot point to be at the 3D Cursor

and then placing the 3D Cursor where you need it. You can either set its coordinates directly

or select a set of vertices or objects in whose center you want the cursor to be and press Shift+S and selecting 'Cursor to Selected'.


Btw: The radius of the sphere is the median distance of the selection to this pivot point.